Scale object to match bounding box python
The bounding box corresponds to a property named 'dimensions'.

So, take the dimensions of each (eventually converted to world coordinates as in your blend file, objects are rotated), calculate the ratio and scale Suzanne by the ratio:

Edit following the comments:

We don't use the world matrix directly, as we only need the rotation part, because the dimensions are already scaled (as said by batFINGER in the comments). So we keep the quaternion.

The abs() is used because the rotations may return a negative value (and we don't want that for dimensions)

Script for Blender 2.79 and below

import bpy

suz = bpy.data.objects['Suzanne']
arm = bpy.data.objects['Armature']

#Convert to world coordinates if you want to use world Y
suzBound = suz.matrix_world.to_quaternion() * suz.dimensions
armBound = arm.matrix_world.to_quaternion() * arm.dimensions

ratio = abs(armBound.y) / abs(suzBound.y)

suz.scale *= ratio


Script for Blender 2.80 and above

import bpy

suz = bpy.data.objects['Suzanne']
arm = bpy.data.objects['Armature']

# Convert to world coordinates if you want to use world Y
# We now use @ instead of * as per https://wiki.blender.org/wiki/Reference/Release_Notes/2.80/Python_API#Matrix_Multiplication
suzBound = suz.matrix_world.to_quaternion() @ suz.dimensions
armBound = arm.matrix_world.to_quaternion() @ arm.dimensions

ratio = abs(armBound.y) / abs(suzBound.y)

suz.scale *= ratio


End of the edit

Just for your information, note that we are talking about bounding boxes here. If the object were rotated differently, that is different than talking about their size along an axis.
For instance, this size: